Dynamics of basket hedging

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Dynamics of basket hedging
(CreditMetrics for baskets the Black-Scholes
of the Credit Derivatives market) Galin
Georgiev January, 2000
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Disclaimer
This report represents only the personal opinions
of the author and not those of J.P.Morgan, its
subsidiaries or affiliates
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Summary

• Definition of a protection contract on an
  individual name and a first-to-default (FTD)
  protection contract on two names
• The CreditMetrics model for baskets basic
  definitions and variations. The basket as a
  rainbow digital option.
• Greeks and dynamic hedging of baskets. Implied
  vs. realized correlation.

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Individual protection contract

• Suppose risk-free interest rates are zero and
  denote by t the present time.
• A protection contract maturing at
  time T on company A entitles the holder to
  receive 1 at T if A defaults prior to T (and 0
  otherwise).
• (In the market place, this is called a
  zero-coupon credit swap, settled at maturity,
  with zero recovery).
• Note that where the latter is the risk-free
  probability of default of A up to time T, i.e.,
  is proportional to the credit
  spread of A.

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FTD protection contract

• A first-to-default (FTD) protection contract
  maturing at time T, on companies A and B,
  entitles the holder to receive 1 at T if at
  least one of the companies defaults prior to T
  (and 0 otherwise).
• The price equals the
  probability of A or B or
  both defaulting before T.

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The market perspective

• While the price of protection for individual
  credits is (more or less) given by the credit
  swaps market, there is no liquid market yet for
  FTD protection (or, equivalently, FTD
  probability).
• One needs a model to price as a
  rainbow derivative on and It depends
  on the correlation between the underlying
  spreads in the no-default state and the
  correlation between the corresponding default
  events.

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The CreditMetrics formalism

• Assumption
• where is a univariate random normal
  variable and is the so-called threshold
  (defined above is the cumulative normal
  distribution).
• One can be more specific and define
• where is the normally distributed firms
  asset level and is the (fixed) firms
  liability level (at time T).

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Asset Distribution at Maturity
Default Probability
Initial Asset Level
Liability Level
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Inconsistencies of the CreditMetrics model
Asset Level
Assets Default
Liability Level 1
Liability Level 2
Assets Liabilities No Default
Time
0
T1
T2
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• Assuming for simplicity constant volatility (of
  the assets), one can rephrase the price of
  protection in terms of familiar option theory
• where is a standard Brownian motion
  which we call normalized threshold (with initial
  point , depending unfortunately on
  T). This is nothing else but the price of a
  barrier option (digital) on the underlying
  struck at 0 and expiring at T.

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The protection contract as a barrier option

• We can therefore think of the protection contract
  as a contingent claim (barrier
  option) on the underlying
• Unsurprisingly, it satisfies the (normal version
  of) the Black-Scholes equation

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The FTD protection contract as a rainbow barrier
option

• The FTD protection price in this context is
• which in terms of normalized thresholds means
• (where is the bivariate normal
  cumulative and is the correlation between
  and ).
• For , one has

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Black-Scholes for the FTD protection

• One can easily compute the Greeks and check that
  our rainbow contingent claim satisfies the
  two-dimensional version of the (normal)
  Black-Scholes equation

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Hedge ratios

• Since the normalized thresholds are not traded,
  we obviously hedge the FTD protection (the
  rainbow barrier option) with the two
  individual protection contracts (1-dim barrier
  options) and . The
  corresponding hedge ratios are
  easily computable

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Convexity of the hedged portfolio

• The hedged FTD portfolio
• is easily seen to have a negative off-diagonal
  convexity
• and positive diagonal convexity ( )

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Convexity seen through the effect of individual
tweaks or parallel tweaks on the hedge ratios of
5 name basket
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Implied vs. realized correlation

• If one buys FTD protection and continuously
  rehedges, the resulting PL is
• where is the realized correlation and
  is the bivariate normal density. If
  , one is long convexity and makes money due to
  rehedging (but one pays for it upfront because
  the money earned by selling the original hedges
  is less).

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A correlation contract ?

• The PL due to continuous rehedging of the basket
  is clearly path-dependent. Similarly to the
  development of the vol contract in standard
  option theory, the time will come to develop a
  correlation contract whose payoff is
  path-independent and proportional to realized
  correlation.